Isotopy-invariant topological measures on closed orientable surfaces of higher genus

Abstract

Given a closed orientable surface of genus at least two, we establish an affine isomorphism between the convex compact set of isotopy-invariant topological measures on and the convex compact set of additive functions on the set of isotopy classes of certain subsurfaces of . We then construct such additive functions, and thus isotopy-invariant topological measures, from probability measures on together with some additional data. The map associating topological measures to probability measures is affine and continuous. Certain Dirac measures map to simple topological measures, while the topological measures due to Py and Rosenberg arise from the normalized Euler characteristic.

Poisson brackets, quasi-states and symplectic integrators

“…Let us also note that in dimension dim M = 2 there exist alternative constructions of symplectic quasi-states (see e.g. [1], [4], [19], [20], [17]) which do not involve Floer homology. None of those quasi-states is known to be induced by a stable quasi-morphism.…”

“…For instance, it is shown in [17] that Py's quasi-morphism [15] gives rise to a quasi-state, but it is unknown whether this quasi-morphism is stable. On the other hand, Zapolsky [19,20] proved that for a wide class of quasi-states ζ on surfaces one has inequality…”

“…In particular, any symplectic quasi-state extends to a topological quasi-state in the sense of Aarnes [1] on the space of continuous functions C(M): This means that it is linear on any singlygenerated subalgebra of C(M). For further discussion on quasi-morphisms and quasi-states on symplectic manifolds, see [4,6,8,9,19,20].…”

Quasi-linear functionals on locally compact spaces

“…Thus, when X is compact, a finite topological measure (and more generally, a bounded signed topological measure) can be defined by its actions on open (respectively, on closed) sets. The idea of determining a topological measure on a closed manifold by its values on closed submanifolds with boundary is in [48,Sect. 2].…”